Question

The matrix $R\left(t\right)$ is defined by $R\left(t\right)=\left[\begin{array}{cc}\cos t& \sin t\\ -\sin t& \cos t\end{array}\right]$. Find $R\left(x\right)R\left(t\right)$.

• A. $R\left(x\right)+R\left(t\right)$
• B. $R\left(x\right)-R\left(t\right)$
• C. $R(x+t)$
• D. None of these.

Answer 1

Answer: (C)

$R(x+t)$

$R\left(x\right)R\left(t\right)=\left[\begin{array}{cc}\cos x& \sin x\\ -\sin x& \cos x\end{array}\right]\left[\begin{array}{cc}\cos t& \sin t\\ -\sin t& \cos t\end{array}\right]$
$\left[\begin{array}{cc}\cos x\cos t-\sin x\sin t& \cos x\sin t+\sin x\cos t\\ -\sin x\cos t-\cos x\sin t& \cos x\cos t-\sin x\sin t\end{array}\right]$
$=\left[\begin{array}{cc}\cos (x+t)& \sin (x+t)\\ -\sin (x+t)& \cos (x+t)\end{array}\right]$ $=R(x+t)$